Turbine nozzle airfoil profile

ABSTRACT

The present application provides a turbine nozzle including an airfoil shape. The airfoil shape may have a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z. The airfoil profile sections at Z distances may be joined smoothly with one another to form a complete airfoil shape.

FIELD

The present application relates generally to a turbine nozzle for a gas turbine engine, and more particularly relates to a turbine nozzle airfoil profile for a turbine nozzle.

BACKGROUND

Some aircraft and/or power plant systems, for example certain jet aircraft, nuclear, simple cycle and combined cycle power plant systems, employ turbines (also referred to as turbomachines) in their design and operation. Some of these turbines employ airfoils (e.g., stator vanes or nozzles and rotor blades.) which during operation are exposed to fluid flows. These airfoils are configured to aerodynamically interact with the fluid flows and generate energy (e.g., creating thrust, turning kinetic energy to mechanical energy, thermal energy to mechanical energy, etc.) from these fluid flows as part of power generation. As a result of this interaction and conversion, the aerodynamic characteristics and losses of these airfoils have an impact on system and turbine operation, performance, thrust, efficiency, and power.

BRIEF DESCRIPTION

Aspects and advantages are set forth below in the following description, or may be obvious from the description, or may be learned through practice.

The present application provides a turbine nozzle including an airfoil shape. The airfoil shape may have a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z. The airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.

The present application further provides a turbine nozzle including an airfoil having a suction-side uncoated nominal airfoil profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance. The airfoil profile sections at the Z distances may be joined smoothly with one another to form a complete suction-side airfoil shape. The X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.

The present application further provides a turbine with a number of turbine nozzles having an airfoil having an airfoil shape. The airfoils having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I. The Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches. The X and Y values, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance. The airfoil profile sections at the Z distances may be joined smoothly with one another to form a complete airfoil shape.

Those of ordinary skill in the art will better appreciate the features and aspects of such embodiments, and others, upon review of the specification.

BRIEF DESCRIPTION OF THE DRAWINGS

A full and enabling disclosure of various embodiments, including the best mode thereof to one skilled in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying figures, in which:

FIG. 1 is a schematic representation of an exemplary turbine having multiple stages with each stage including alternating rows of turbine rotor blades and turbine nozzles according to at least one embodiment of the present disclosure;

FIG. 2 is a perspective view of an exemplary turbine nozzle according to at least one embodiment of the present disclosure; and

FIG. 3 provides a cross-sectional view of an airfoil portion of the turbine nozzle of FIG. 2.

DETAILED DESCRIPTION

Referring now to the drawings, particularly to FIG. 1, there is illustrated an exemplary turbine 10 of a gas turbine 12 including a plurality of turbine stages arranged in serial flow order. Each stage of the turbine includes a row of turbine nozzles or vanes disposed axially adjacent to a corresponding row of turbine rotor blades. Four stages are illustrated in FIG. 1. The exact number of stages of the turbine 10 is a choice of engineering design, and may be more or less than the four stages illustrated in FIG. 1. The four stages are merely exemplary of one turbine design, and are not intended to limit the presently claimed turbine rotor blade in any manner.

The first stage comprises a plurality of turbine nozzles or vanes 100 and a plurality of turbine rotor blades 150. The turbine nozzles 100 are annularly arranged about an axis of a turbine rotor 14. The turbine rotor blades 150 are annularly arranged about and coupled to the turbine rotor 14. A second stage of the turbine 12 includes a plurality of turbine nozzles 200 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 250 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 200 may be coupled to a casing 16 of the turbine 12.

As shown in FIG. 1, a third stage of the turbine 12 includes a plurality of turbine nozzles 300 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 350 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 300 may be coupled to the casing 16 of the turbine 12. In particular embodiments, the turbine 12 includes a fourth stage which includes plurality of turbine nozzles 400 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 450 annularly arranged about and coupled to the turbine rotor 14. The fourth stage turbine nozzles 400 may be coupled to the casing 16 of the turbine 12.

It will be appreciated that the turbine nozzles 100, 200, 300 and 400 and turbine rotor blades 150, 250, 350 and 450 are disposed or at least partially disposed within a hot gas path 18 of the turbine 12. The various stages of the turbine 10 at least partially define the hot gas path 18 through which combustion gases, as indicated by arrows 20, flow during operation of the gas turbine 12.

FIG. 2 provides a perspective view of an exemplary turbine nozzle 50 as may be incorporated in any stage of the turbine 12. FIG. 3 provides a cross-sectional top view of an airfoil 52 of the turbine nozzle 50 shown in FIG. 2. In one embodiment the turbine nozzle 50 as provided in Table I below is representative of turbine nozzle 100 of the plurality of turbine nozzles 100 of the first stage as shown in FIG. 1. As shown in FIGS. 2 and 3 collectively, the turbine nozzle 50 airfoil 52 includes a pressure-side wall 54 and an opposing suction-side wall 56. The pressure-side wall 54 and the suction-side wall 56 meet or intersect at a leading edge 58 and at a trailing edge 60 of the airfoil 52. A chord line 62 extends between the leading edge 58 and the trailing edge 60 such that pressure and suction-side walls 54, 56 can be said to extend in chord or chordwise between the leading edge 58 and the trailing edge 60.

As shown in FIG. 2, the airfoil 52 includes a root or first end 64 which intersects with and extends radially outwardly from an inner band 66 of the turbine nozzle 50. The airfoil 52 terminates radially at a second end or tip 68 of the airfoil 52. The second end 68 intersects with and extends radially inwardly from an outer band 70 of the turbine nozzle 50. The pressure and suction-side walls 54, 56 can be said to extend in height and/or span or in a span-wise direction 72 between the root 64 and/or the inner band 66 and the second end 68 or the outer band 70. In other words, each turbine nozzle 50 includes an airfoil 52 having opposing pressure and suction-side walls 54, 56 that extend in chord or chordwise 62 between opposing leading and trailing edges 58, 60 and that extend in span or span-wise 72 between the root 64 and the second end 68 of the airfoil 52.

In particular configurations, the airfoil 52 may include a fillet 74 formed between the inner band 66 and the airfoil 52 proximate to the root 64. The fillet 74 can include a weld or braze fillet, which can be formed via conventional MIG welding, TIG welding, brazing, etc., and can include a profile that can reduce fluid dynamic losses as a result of the presence of fillet 74. In particular embodiments, the inner band 66, airfoil 52 and the fillet 74 can be formed as a single component, such as by casting and/or machining and/or 3D printing and/or any other suitable technique now known or later developed and/or discovered.

The airfoil 52 of the turbine nozzle 50 has a profile at any cross-section taken between the inner band 66 or the root 64 and the second end 68. The X, Y, and Z values are given in Table I as percentage values of the airfoil length or height/span. As one example only, the height or radial span of the airfoil 52 of turbine nozzle 50 may be from about 5 inches to about 50 inches (about 12 centimeters to about 130 centimeters). However, it is to be understood that heights below or above this range may also be employed as desired in the specific application.

The gas turbine hot gas path 18 requires airfoils 52 that meet system requirements of aerodynamic and mechanical blade loading and efficiency. To define the airfoil shape of each turbine nozzle airfoil 52, there is a unique set or loci of points in space that meet the stage requirements and can be manufactured. These unique loci of points meet the requirements for stage efficiency and are arrived at by iteration between aerodynamic and mechanical loadings enabling the turbine to run in an efficient, safe and smooth manner. These points are unique and specific to the system.

The loci that defines the turbine nozzle airfoil profile includes a set of about 2,200 points with X, Y and Z dimensions relative to a reference origin coordinate system. The Cartesian coordinate system of X, Y and Z values given in Table I below defines the profile of the turbine nozzle airfoil at various locations along its length. Table I lists data for a non-coated airfoil. The envelope/tolerance for the coordinates is about +/−5% in a direction normal to any airfoil surface location and/or about +/−5% of the chord 62 in a direction nominal to any airfoil surface location. The point data origin 76 is defined at or proximate to the leading edge 58 of the airfoil 52 and at or proximate to the root 64.

The coordinate values for the X, Y and Z coordinates are set forth in non-dimensionalized units by the values provided in Table I although other units of dimensions may be used when the values are appropriately converted. The X, Y and Z values set forth in Table I are also expressed in non-dimensional form (X, Y, and Z) from 0% to 100% of the height or span of the airfoil 52. As one example only, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the leading edge 58 and multiplying by a constant number (e.g., 100). As one example only, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the trailing edge 60 and multiplying by a constant number (e.g., 100).

To convert the Z value to a Z coordinate value, e.g., in inches, the non-dimensional Z value given in Table I is multiplied by the Z length of the airfoil in inches. As described above, the Cartesian coordinate system has orthogonally-related X, Y and Z axes and the X axis lies generally parallel to the turbine rotor centerline, i.e., the rotary axis and a positive X coordinate value is axial toward an exhaust end of the turbine 14. The positive Y coordinate value extends tangentially in the direction of rotation of the rotor shaft 14 and the positive Z coordinate value is radially outwardly toward the second end 68 of the airfoil 52. All the values in Table I are given at room temperature and are non-filleted.

By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the profile section or airfoil shape of the airfoil 52 of the turbine nozzle 50, at each Z distance along the length of the airfoil 52 can be ascertained. By connecting the X and Y values with smooth continuing arcs, each profile section at each distance Z is fixed. The airfoil profiles of the various surface locations between the distances Z are determined by smoothly connecting the adjacent profile sections to one another to form the airfoil profile.

The Table I values are generated and shown to four decimal places for determining the profile of the airfoil. As the turbine nozzle surface heats up during operation of the turbine 14, mechanical stresses and elevated temperatures will cause a change in the X, Y and Z values. Accordingly, the values for the profile given in Table I represent ambient, non-operating or non-hot conditions (e.g., room temperature) and are for an uncoated airfoil.

There are typical manufacturing tolerances as well as coatings which must be accounted for in the actual profile of the airfoil 52. Each section is joined smoothly with the other sections to form the complete airfoil shape. It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/− values, including any coating thicknesses, are additive to the X and Y values given in Table I below. Accordingly, a distance of +/−5% in a direction normal to any surface location along the airfoil profile defines an airfoil profile envelope for this particular turbine nozzle airfoil design and turbine, i.e., a range of variation between measured points on the actual airfoil surface at nominal cold or room temperature and the ideal position of those points as given in the Table below at the same temperature. The data is scalable and the geometry pertains to all aerodynamic scales, at above and/or below 3000 RPM. The design of the airfoil 52 of the turbine nozzle 50 is robust to this range of variation without impairment of mechanical and aerodynamic functions.

TABLE I SUCTION-SIDE PRESSURE-SIDE X (%) Y (%) Z (%) N X (%) Y (%) Z (%) 0.0000 0.0000 0.0000 1 0.0000 0.0000 0.0000 0.4049 −1.2106 0.0000 2 −0.1757 1.0178 0.0000 0.9846 −2.3483 0.0000 3 −0.2026 2.0503 0.0000 1.7039 −3.4036 0.0000 4 −0.0786 3.0757 0.0000 2.5315 −4.3766 0.0000 5 0.1873 4.0738 0.0000 3.4429 −5.2717 0.0000 6 0.5791 5.0297 0.0000 4.4206 −6.0941 0.0000 7 1.0770 5.9352 0.0000 5.4507 −6.8500 0.0000 8 1.6629 6.7863 0.0000 6.5213 −7.5474 0.0000 9 2.3203 7.5838 0.0000 7.6260 −8.1895 0.0000 10 3.0364 8.3290 0.0000 8.7642 −8.7700 0.0000 11 3.8001 9.0255 0.0000 9.9360 −9.2790 0.0000 12 4.5993 9.6811 0.0000 11.1410 −9.7032 0.0000 13 5.4137 10.3178 0.0000 12.3759 −10.0305 0.0000 14 6.2239 10.9598 0.0000 13.6343 −10.2503 0.0000 15 7.0279 11.6096 0.0000 14.9075 −10.3539 0.0000 16 7.8290 12.2630 0.0000 16.1847 −10.3363 0.0000 17 8.6270 12.9200 0.0000 17.4542 −10.1949 0.0000 18 9.4213 13.5816 0.0000 18.7049 −9.9349 0.0000 19 10.2115 14.2481 0.0000 19.9273 −9.5639 0.0000 20 10.9980 14.9191 0.0000 21.1143 −9.0915 0.0000 21 11.7808 15.5942 0.0000 22.2619 −8.5302 0.0000 22 12.5598 16.2737 0.0000 23.3683 −7.8913 0.0000 23 13.3349 16.9578 0.0000 24.4332 −7.1853 0.0000 24 14.1059 17.6464 0.0000 25.4578 −6.4219 0.0000 25 14.8729 18.3394 0.0000 26.4440 −5.6095 0.0000 26 15.6361 19.0367 0.0000 27.3939 −4.7548 0.0000 27 16.3951 19.7385 0.0000 28.3097 −3.8637 0.0000 28 17.1500 20.4447 0.0000 29.1940 −2.9413 0.0000 29 17.9007 21.1555 0.0000 30.0489 −1.9916 0.0000 30 18.6471 21.8706 0.0000 30.8765 −1.0181 0.0000 31 19.3894 22.5901 0.0000 31.6791 −0.0237 0.0000 32 20.1275 23.3139 0.0000 32.4584 0.9890 0.0000 33 20.8614 24.0419 0.0000 33.2160 2.0180 0.0000 34 21.5908 24.7744 0.0000 33.9534 3.0616 0.0000 35 22.3157 25.5114 0.0000 34.6721 4.1182 0.0000 36 23.0362 26.2527 0.0000 35.3733 5.1865 0.0000 37 23.7521 26.9985 0.0000 36.0581 6.2653 0.0000 38 24.4635 27.7485 0.0000 36.7277 7.3537 0.0000 39 25.1704 28.5028 0.0000 37.3831 8.4507 0.0000 40 25.8727 29.2613 0.0000 38.0248 9.5557 0.0000 41 26.5704 30.0241 0.0000 38.6538 10.6681 0.0000 42 27.2634 30.7912 0.0000 39.2707 11.7872 0.0000 43 27.9516 31.5625 0.0000 39.8758 12.9126 0.0000 44 28.6351 32.3380 0.0000 40.4694 14.0443 0.0000 45 29.3138 33.1178 0.0000 41.0516 15.1818 0.0000 46 29.9876 33.9018 0.0000 41.6226 16.3250 0.0000 47 30.6566 34.6899 0.0000 42.1825 17.4736 0.0000 48 31.3206 35.4822 0.0000 42.7315 18.6275 0.0000 49 31.9797 36.2785 0.0000 43.2698 19.7865 0.0000 50 32.6338 37.0790 0.0000 43.7977 20.9502 0.0000 51 33.2829 37.8836 0.0000 44.3153 22.1185 0.0000 52 33.9269 38.6922 0.0000 44.8228 23.2913 0.0000 53 34.5657 39.5050 0.0000 45.3203 24.4683 0.0000 54 35.1994 40.3217 0.0000 45.8080 25.6494 0.0000 55 35.8278 41.1425 0.0000 46.2861 26.8345 0.0000 56 36.4510 41.9673 0.0000 46.7548 28.0233 0.0000 57 37.0689 42.7960 0.0000 47.2142 29.2157 0.0000 58 37.6816 43.6287 0.0000 47.6644 30.4116 0.0000 59 38.2889 44.4652 0.0000 48.1058 31.6108 0.0000 60 38.8909 45.3056 0.0000 48.5384 32.8132 0.0000 61 39.4875 46.1498 0.0000 48.9624 34.0187 0.0000 62 40.0787 46.9978 0.0000 49.3780 35.2271 0.0000 63 40.6644 47.8496 0.0000 49.7852 36.4383 0.0000 64 41.2446 48.7052 0.0000 50.1842 37.6523 0.0000 65 41.8193 49.5645 0.0000 50.5752 38.8689 0.0000 66 42.3884 50.4274 0.0000 50.9583 40.0879 0.0000 67 42.9520 51.2940 0.0000 51.3335 41.3095 0.0000 68 43.5100 52.1643 0.0000 51.7012 42.5333 0.0000 69 44.0623 53.0381 0.0000 52.0613 43.7594 0.0000 70 44.6089 53.9155 0.0000 52.4141 44.9875 0.0000 71 45.1499 54.7964 0.0000 52.7596 46.2178 0.0000 72 45.6852 55.6808 0.0000 53.0980 47.4500 0.0000 73 46.2147 56.5686 0.0000 53.4294 48.6842 0.0000 74 46.7385 57.4598 0.0000 53.7540 49.9201 0.0000 75 47.2566 58.3543 0.0000 54.0718 51.1578 0.0000 76 47.7689 59.2522 0.0000 54.3830 52.3972 0.0000 77 48.2754 60.1534 0.0000 54.6876 53.6382 0.0000 78 48.7761 61.0578 0.0000 54.9859 54.8808 0.0000 79 49.2710 61.9654 0.0000 55.2779 56.1248 0.0000 80 49.7600 62.8762 0.0000 55.5637 57.3703 0.0000 81 50.2431 63.7901 0.0000 55.8434 58.6172 0.0000 82 50.7203 64.7070 0.0000 56.1173 59.8654 0.0000 83 51.1917 65.6271 0.0000 56.3854 61.1148 0.0000 84 51.6572 66.5501 0.0000 56.6477 62.3654 0.0000 85 52.1167 67.4761 0.0000 56.9045 63.6172 0.0000 86 52.5703 68.4050 0.0000 57.1557 64.8701 0.0000 87 53.0179 69.3368 0.0000 57.4016 66.1241 0.0000 88 53.4594 70.2715 0.0000 57.6420 67.3791 0.0000 89 53.8947 71.2091 0.0000 57.8773 68.6352 0.0000 90 54.3240 72.1496 0.0000 58.1074 69.8921 0.0000 91 54.7471 73.0928 0.0000 58.3325 71.1500 0.0000 92 55.1640 74.0387 0.0000 58.5529 72.4087 0.0000 93 55.5750 74.9872 0.0000 58.7687 73.6682 0.0000 94 55.9801 75.9383 0.0000 58.9798 74.9285 0.0000 95 56.3790 76.8920 0.0000 59.1865 76.1895 0.0000 96 56.7720 77.8481 0.0000 59.3886 77.4513 0.0000 97 57.1585 78.8069 0.0000 59.5862 78.7138 0.0000 98 57.5390 79.7681 0.0000 59.7428 79.9789 0.0000 99 58.0001 80.6820 0.0000 58.9641 80.9009 0.0000 100 58.9641 80.9009 0.0000 0.0500 −1.7399 10.0000 1 0.0500 −1.7399 10.0000 0.6620 −2.8499 10.0000 2 −0.3306 −0.7607 10.0000 1.4064 −3.8761 10.0000 3 −0.5737 0.2611 10.0000 2.2534 −4.8196 10.0000 4 −0.6628 1.3076 10.0000 3.1797 −5.6857 10.0000 5 −0.5907 2.3554 10.0000 4.1678 −6.4806 10.0000 6 −0.3631 3.3808 10.0000 5.2048 −7.2107 10.0000 7 0.0036 4.3653 10.0000 6.2805 −7.8826 10.0000 8 0.4885 5.2974 10.0000 7.3901 −8.4969 10.0000 9 1.0706 6.1723 10.0000 8.5330 −9.0466 10.0000 10 1.7308 6.9900 10.0000 9.7083 −9.5229 10.0000 11 2.4549 7.7518 10.0000 10.9143 −9.9152 10.0000 12 3.2313 8.4604 10.0000 12.1467 −10.2140 10.0000 13 4.0421 9.1295 10.0000 13.3995 −10.4099 10.0000 14 4.8684 9.7795 10.0000 14.6646 −10.4960 10.0000 15 5.6904 10.4349 10.0000 15.9323 −10.4687 10.0000 16 6.5055 11.0988 10.0000 17.1923 −10.3267 10.0000 17 7.3160 11.7684 10.0000 18.4350 −10.0745 10.0000 18 8.1227 12.4424 10.0000 19.6519 −9.7180 10.0000 19 8.9253 13.1215 10.0000 20.8365 −9.2652 10.0000 20 9.7230 13.8063 10.0000 21.9847 −8.7268 10.0000 21 10.5159 14.4966 10.0000 23.0942 −8.1125 10.0000 22 11.3044 15.1918 10.0000 24.1642 −7.4317 10.0000 23 12.0886 15.8921 10.0000 25.1952 −6.6931 10.0000 24 12.8679 16.5977 10.0000 26.1886 −5.9045 10.0000 25 13.6425 17.3085 10.0000 27.1457 −5.0723 10.0000 26 14.4125 18.0242 10.0000 28.0686 −4.2022 10.0000 27 15.1780 18.7448 10.0000 28.9592 −3.2992 10.0000 28 15.9388 19.4703 10.0000 29.8196 −2.3672 10.0000 29 16.6947 20.2009 10.0000 30.6514 −1.4096 10.0000 30 17.4459 20.9364 10.0000 31.4565 −0.4296 10.0000 31 18.1923 21.6768 10.0000 32.2368 0.5704 10.0000 32 18.9339 22.4218 10.0000 32.9937 1.5883 10.0000 33 19.6709 23.1715 10.0000 33.7285 2.6221 10.0000 34 20.4031 23.9259 10.0000 34.4429 3.6702 10.0000 35 21.1304 24.6851 10.0000 35.1380 4.7312 10.0000 36 21.8527 25.4489 10.0000 35.8150 5.8039 10.0000 37 22.5700 26.2175 10.0000 36.4749 6.8871 10.0000 38 23.2825 26.9905 10.0000 37.1187 7.9800 10.0000 39 23.9900 27.7681 10.0000 37.7472 9.0818 10.0000 40 24.6927 28.5501 10.0000 38.3609 10.1918 10.0000 41 25.3903 29.3365 10.0000 38.9606 11.3096 10.0000 42 26.0830 30.1274 10.0000 39.5466 12.4345 10.0000 43 26.7705 30.9227 10.0000 40.1194 13.5662 10.0000 44 27.4530 31.7223 10.0000 40.6792 14.7044 10.0000 45 28.1305 32.5261 10.0000 41.2263 15.8488 10.0000 46 28.8030 33.3342 10.0000 41.7611 16.9990 10.0000 47 29.4704 34.1465 10.0000 42.2838 18.1547 10.0000 48 30.1328 34.9629 10.0000 42.7949 19.3156 10.0000 49 30.7900 35.7834 10.0000 43.2946 20.4814 10.0000 50 31.4420 36.6081 10.0000 43.7833 21.6519 10.0000 51 32.0889 37.4368 10.0000 44.2612 22.8269 10.0000 52 32.7306 38.2695 10.0000 44.7287 24.0060 10.0000 53 33.3670 39.1063 10.0000 45.1860 25.1892 10.0000 54 33.9982 39.9470 10.0000 45.6334 26.3761 10.0000 55 34.6242 40.7916 10.0000 46.0711 27.5666 10.0000 56 35.2449 41.6401 10.0000 46.4995 28.7605 10.0000 57 35.8602 42.4925 10.0000 46.9187 29.9576 10.0000 58 36.4702 43.3487 10.0000 47.3291 31.1578 10.0000 59 37.0749 44.2087 10.0000 47.7309 32.3609 10.0000 60 37.6743 45.0724 10.0000 48.1244 33.5668 10.0000 61 38.2683 45.9398 10.0000 48.5097 34.7753 10.0000 62 38.8569 46.8108 10.0000 48.8870 35.9863 10.0000 63 39.4402 47.6855 10.0000 49.2567 37.1996 10.0000 64 40.0181 48.5637 10.0000 49.6187 38.4153 10.0000 65 40.5905 49.4455 10.0000 49.9734 39.6331 10.0000 66 41.1576 50.3307 10.0000 50.3210 40.8530 10.0000 67 41.7191 51.2195 10.0000 50.6617 42.0748 10.0000 68 42.2751 52.1117 10.0000 50.9956 43.2985 10.0000 69 42.8256 53.0073 10.0000 51.3229 44.5240 10.0000 70 43.3706 53.9063 10.0000 51.6438 45.7511 10.0000 71 43.9101 54.8087 10.0000 51.9585 46.9799 10.0000 72 44.4440 55.7143 10.0000 52.2672 48.2102 10.0000 73 44.9723 56.6231 10.0000 52.5699 49.4420 10.0000 74 45.4951 57.5352 10.0000 52.8668 50.6752 10.0000 75 46.0123 58.4505 10.0000 53.1581 51.9097 10.0000 76 46.5239 59.3689 10.0000 53.4439 53.1455 10.0000 77 47.0299 60.2905 10.0000 53.7244 54.3826 10.0000 78 47.5302 61.2151 10.0000 53.9997 55.6207 10.0000 79 48.0248 62.1427 10.0000 54.2699 56.8601 10.0000 80 48.5138 63.0734 10.0000 54.5352 58.1004 10.0000 81 48.9970 64.0070 10.0000 54.7958 59.3418 10.0000 82 49.4745 64.9436 10.0000 55.0516 60.5842 10.0000 83 49.9464 65.8831 10.0000 55.3029 61.8275 10.0000 84 50.4125 66.8254 10.0000 55.5498 63.0716 10.0000 85 50.8728 67.7705 10.0000 55.7923 64.3167 10.0000 86 51.3274 68.7185 10.0000 56.0307 65.5625 10.0000 87 51.7762 69.6691 10.0000 56.2648 66.8091 10.0000 88 52.2191 70.6226 10.0000 56.4950 68.0565 10.0000 89 52.6561 71.5788 10.0000 56.7212 69.3046 10.0000 90 53.0871 72.5376 10.0000 56.9435 70.5534 10.0000 91 53.5123 73.4991 10.0000 57.1623 71.8028 10.0000 92 53.9315 74.4632 10.0000 57.3774 73.0529 10.0000 93 54.3452 75.4297 10.0000 57.5892 74.3035 10.0000 94 54.7531 76.3986 10.0000 57.7976 75.5547 10.0000 95 55.1551 77.3700 10.0000 58.0027 76.8064 10.0000 96 55.5513 78.3437 10.0000 58.2045 78.0587 10.0000 97 55.9418 79.3198 10.0000 58.4031 79.3115 10.0000 98 56.3251 80.2988 10.0000 58.5593 80.5669 10.0000 99 56.7988 81.2258 10.0000 57.7762 81.4697 10.0000 100 57.7762 81.4697 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7.0694 80.0000 9.3519 −11.5509 80.0000 13 2.9846 7.7876 80.0000 10.7057 −11.6832 80.0000 14 3.8281 8.5119 80.0000 12.0655 −11.7149 80.0000 15 4.6659 9.2427 80.0000 13.4240 −11.6475 80.0000 16 5.4970 9.9812 80.0000 14.7745 −11.4844 80.0000 17 6.3217 10.7268 80.0000 16.1106 −11.2297 80.0000 18 7.1404 11.4790 80.0000 17.4274 −10.8884 80.0000 19 7.9527 12.2380 80.0000 18.7204 −10.4657 80.0000 20 8.7585 13.0040 80.0000 19.9860 −9.9671 80.0000 21 9.5578 13.7768 80.0000 21.2217 −9.3982 80.0000 22 10.3507 14.5561 80.0000 22.4253 −8.7642 80.0000 23 11.1375 15.3416 80.0000 23.5954 −8.0703 80.0000 24 11.9184 16.1330 80.0000 24.7313 −7.3217 80.0000 25 12.6932 16.9303 80.0000 25.8325 −6.5229 80.0000 26 13.4621 17.7334 80.0000 26.8988 −5.6781 80.0000 27 14.2250 18.5421 80.0000 27.9302 −4.7911 80.0000 28 14.9818 19.3565 80.0000 28.9272 −3.8655 80.0000 29 15.7328 20.1763 80.0000 29.8900 −2.9043 80.0000 30 16.4779 21.0014 80.0000 30.8193 −1.9107 80.0000 31 17.2174 21.8316 80.0000 31.7156 −0.8872 80.0000 32 17.9513 22.6668 80.0000 32.5796 0.1636 80.0000 33 18.6795 23.5069 80.0000 33.4119 1.2398 80.0000 34 19.4021 24.3518 80.0000 34.2131 2.3393 80.0000 35 20.1191 25.2015 80.0000 34.9840 3.4603 80.0000 36 20.8304 26.0559 80.0000 35.7250 4.6012 80.0000 37 21.5362 26.9150 80.0000 36.4371 5.7605 80.0000 38 22.2363 27.7786 80.0000 37.1208 6.9366 80.0000 39 22.9309 28.6467 80.0000 37.7775 8.1281 80.0000 40 23.6200 29.5192 80.0000 38.4085 9.3335 80.0000 41 24.3038 30.3958 80.0000 39.0152 10.5512 80.0000 42 24.9822 31.2766 80.0000 39.5993 11.7799 80.0000 43 25.6554 32.1614 80.0000 40.1620 13.0186 80.0000 44 26.3233 33.0502 80.0000 40.7046 14.2662 80.0000 45 26.9860 33.9429 80.0000 41.2284 15.5218 80.0000 46 27.6434 34.8395 80.0000 41.7346 16.7846 80.0000 47 28.2957 35.7398 80.0000 42.2244 18.0539 80.0000 48 28.9429 36.6438 80.0000 42.6986 19.3291 80.0000 49 29.5851 37.5514 80.0000 43.1584 20.6095 80.0000 50 30.2222 38.4625 80.0000 43.6046 21.8948 80.0000 51 30.8543 39.3770 80.0000 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42.4942 58.3121 80.0000 50.9241 49.4959 80.0000 72 43.0284 59.2871 80.0000 51.2097 50.8261 80.0000 73 43.5582 60.2646 80.0000 51.4917 52.1570 80.0000 74 44.0838 61.2443 80.0000 51.7703 53.4887 80.0000 75 44.6050 62.2263 80.0000 52.0455 54.8211 80.0000 76 45.1220 63.2106 80.0000 52.3177 56.1541 80.0000 77 45.6347 64.1971 80.0000 52.5870 57.4877 80.0000 78 46.1432 65.1857 80.0000 52.8534 58.8219 80.0000 79 46.6475 66.1766 80.0000 53.1172 60.1565 80.0000 80 47.1477 67.1695 80.0000 53.3785 61.4917 80.0000 81 47.6437 68.1645 80.0000 53.6373 62.8274 80.0000 82 48.1355 69.1616 80.0000 53.8939 64.1635 80.0000 83 48.6233 70.1606 80.0000 54.1482 65.5000 80.0000 84 49.1070 71.1617 80.0000 54.4005 66.8369 80.0000 85 49.5867 72.1647 80.0000 54.6509 68.1742 80.0000 86 50.0623 73.1696 80.0000 54.8993 69.5118 80.0000 87 50.5340 74.1763 80.0000 55.1460 70.8498 80.0000 88 51.0017 75.1850 80.0000 55.3911 72.1880 80.0000 89 51.4655 76.1954 80.0000 55.6345 73.5266 80.0000 90 51.9254 77.2076 80.0000 55.8765 74.8654 80.0000 91 52.3814 78.2215 80.0000 56.1170 76.2045 80.0000 92 52.8335 79.2372 80.0000 56.3562 77.5438 80.0000 93 53.2818 80.2547 80.0000 56.5941 78.8834 80.0000 94 53.7261 81.2738 80.0000 56.8307 80.2231 80.0000 95 54.1666 82.2946 80.0000 57.0661 81.5631 80.0000 96 54.6033 83.3170 80.0000 57.3005 82.9033 80.0000 97 55.0362 84.3410 80.0000 57.5338 84.2436 80.0000 98 55.4656 85.3665 80.0000 57.7524 85.5857 80.0000 99 55.9685 86.3485 80.0000 56.9985 86.6044 80.0000 100 56.9985 86.6044 80.0000 −3.9489 −3.3922 90.0000 1 −3.9489 −3.3922 90.0000 −3.3747 −4.6318 90.0000 2 −4.2199 −2.3027 90.0000 −2.5974 −5.7562 90.0000 3 −4.2756 −1.1816 90.0000 −1.6748 −6.7656 90.0000 4 −4.1087 −0.0715 90.0000 −0.6493 −7.6705 90.0000 5 −3.7403 0.9892 90.0000 0.4517 −8.4822 90.0000 6 −3.2077 1.9783 90.0000 1.6086 −9.2123 90.0000 7 −2.5486 2.8883 90.0000 2.8097 −9.8673 90.0000 8 −1.7943 3.7214 90.0000 4.0515 −10.4410 90.0000 9 −0.9655 4.4807 90.0000 5.3303 −10.9270 90.0000 10 −0.0972 5.1948 90.0000 6.6405 −11.3203 90.0000 11 0.7801 5.8980 90.0000 7.9757 −11.6176 90.0000 12 1.6498 6.6106 90.0000 9.3291 −11.8170 90.0000 13 2.5089 7.3359 90.0000 10.6933 −11.9175 90.0000 14 3.3618 8.0684 90.0000 12.0613 −11.9191 90.0000 15 4.2093 8.8072 90.0000 13.4259 −11.8239 90.0000 16 5.0501 9.5537 90.0000 14.7809 −11.6353 90.0000 17 5.8843 10.3074 90.0000 16.1204 −11.3576 90.0000 18 6.7126 11.0678 90.0000 17.4398 −10.9960 90.0000 19 7.5345 11.8349 90.0000 18.7351 −10.5558 90.0000 20 8.3497 12.6092 90.0000 20.0031 −10.0422 90.0000 21 9.1583 13.3904 90.0000 21.2412 −9.4601 90.0000 22 9.9604 14.1783 90.0000 22.4473 −8.8144 90.0000 23 10.7563 14.9724 90.0000 23.6203 −8.1102 90.0000 24 11.5461 15.7726 90.0000 24.7593 −7.3523 90.0000 25 12.3298 16.5788 90.0000 25.8639 −6.5450 90.0000 26 13.1074 17.3908 90.0000 26.9338 −5.6923 90.0000 27 13.8791 18.2085 90.0000 27.9692 −4.7980 90.0000 28 14.6447 19.0318 90.0000 28.9703 −3.8653 90.0000 29 15.4044 19.8606 90.0000 29.9373 −2.8974 90.0000 30 16.1582 20.6948 90.0000 30.8705 −1.8969 90.0000 31 16.9062 21.5342 90.0000 31.7707 −0.8665 90.0000 32 17.6485 22.3787 90.0000 32.6381 0.1915 90.0000 33 18.3851 23.2282 90.0000 33.4735 1.2751 90.0000 34 19.1160 24.0824 90.0000 34.2772 2.3824 90.0000 35 19.8414 24.9414 90.0000 35.0500 3.5114 90.0000 36 20.5612 25.8051 90.0000 35.7925 4.6607 90.0000 37 21.2753 26.6735 90.0000 36.5053 5.8285 90.0000 38 21.9838 27.5466 90.0000 37.1892 7.0135 90.0000 39 22.6866 28.4242 90.0000 37.8455 8.2141 90.0000 40 23.3839 29.3061 90.0000 38.4759 9.4285 90.0000 41 24.0759 30.1923 90.0000 39.0819 10.6552 90.0000 42 24.7626 31.0825 90.0000 39.6652 11.8928 90.0000 43 25.4440 31.9768 90.0000 40.2273 13.1403 90.0000 44 26.1202 32.8751 90.0000 40.7695 14.3965 90.0000 45 26.7911 33.7773 90.0000 41.2930 15.6606 90.0000 46 27.4568 34.6833 90.0000 41.7993 16.9318 90.0000 47 28.1174 35.5931 90.0000 42.2893 18.2093 90.0000 48 28.7729 36.5066 90.0000 42.7641 19.4925 90.0000 49 29.4233 37.4237 90.0000 43.2248 20.7809 90.0000 50 30.0687 38.3443 90.0000 43.6722 22.0739 90.0000 51 30.7092 39.2684 90.0000 44.1072 23.3712 90.0000 52 31.3447 40.1958 90.0000 44.5305 24.6722 90.0000 53 31.9754 41.1266 90.0000 44.9429 25.9769 90.0000 54 32.6013 42.0606 90.0000 45.3450 27.2847 90.0000 55 33.2223 42.9978 90.0000 45.7376 28.5954 90.0000 56 33.8386 43.9382 90.0000 46.1210 29.9089 90.0000 57 34.4500 44.8818 90.0000 46.4960 31.2247 90.0000 58 35.0567 45.8283 90.0000 46.8630 32.5429 90.0000 59 35.6587 46.7779 90.0000 47.2225 33.8630 90.0000 60 36.2560 47.7305 90.0000 47.5750 35.1851 90.0000 61 36.8487 48.6859 90.0000 47.9208 36.5089 90.0000 62 37.4368 49.6441 90.0000 48.2604 37.8344 90.0000 63 38.0203 50.6052 90.0000 48.5940 39.1613 90.0000 64 38.5993 51.5689 90.0000 48.9221 40.4897 90.0000 65 39.1739 52.5353 90.0000 49.2450 41.8193 90.0000 66 39.7440 53.5044 90.0000 49.5628 43.1501 90.0000 67 40.3097 54.4760 90.0000 49.8758 44.4821 90.0000 68 40.8711 55.4502 90.0000 50.1845 45.8151 90.0000 69 41.4281 56.4268 90.0000 50.4889 47.1490 90.0000 70 41.9808 57.4059 90.0000 50.7893 48.4839 90.0000 71 42.5292 58.3874 90.0000 51.0859 49.8196 90.0000 72 43.0734 59.3712 90.0000 51.3789 51.1561 90.0000 73 43.6134 60.3574 90.0000 51.6686 52.4933 90.0000 74 44.1492 61.3459 90.0000 51.9550 53.8313 90.0000 75 44.6808 62.3366 90.0000 52.2384 55.1699 90.0000 76 45.2083 63.3295 90.0000 52.5189 56.5091 90.0000 77 45.7317 64.3245 90.0000 52.7966 57.8488 90.0000 78 46.2510 65.3217 90.0000 53.0717 59.1891 90.0000 79 46.7662 66.3211 90.0000 53.3444 60.5299 90.0000 80 47.2775 67.3224 90.0000 53.6148 61.8712 90.0000 81 47.7847 68.3258 90.0000 53.8829 63.2130 90.0000 82 48.2880 69.3312 90.0000 54.1489 64.5551 90.0000 83 48.7874 70.3385 90.0000 54.4129 65.8976 90.0000 84 49.2829 71.3478 90.0000 54.6750 67.2406 90.0000 85 49.7744 72.3589 90.0000 54.9354 68.5838 90.0000 86 50.2622 73.3719 90.0000 55.1941 69.9274 90.0000 87 50.7462 74.3868 90.0000 55.4512 71.2713 90.0000 88 51.2264 75.4034 90.0000 55.7068 72.6154 90.0000 89 51.7030 76.4217 90.0000 55.9610 73.9599 90.0000 90 52.1759 77.4417 90.0000 56.2138 75.3046 90.0000 91 52.6451 78.4634 90.0000 56.4655 76.6495 90.0000 92 53.1107 79.4868 90.0000 56.7159 77.9946 90.0000 93 53.5727 80.5119 90.0000 56.9653 79.3399 90.0000 94 54.0310 81.5385 90.0000 57.2136 80.6855 90.0000 95 54.4856 82.5668 90.0000 57.4609 82.0312 90.0000 96 54.9368 83.5967 90.0000 57.7074 83.3771 90.0000 97 55.3845 84.6280 90.0000 57.9531 84.7231 90.0000 98 55.8289 85.6608 90.0000 58.1649 86.0723 90.0000 99 56.3122 86.6717 90.0000 57.3240 87.0271 90.0000 100 57.3240 87.0271 90.0000 −4.5483 −3.2441 100.0000 1 −4.5483 −3.2441 100.0000 −4.0492 −4.5394 100.0000 2 −4.7268 −2.1205 100.0000 −3.3137 −5.7178 100.0000 3 −4.6714 −0.9844 100.0000 −2.4082 −6.7720 100.0000 4 −4.3899 0.1181 100.0000 −1.3856 −7.7137 100.0000 5 −3.9165 1.1533 100.0000 −0.2787 −8.5551 100.0000 6 −3.2946 2.1073 100.0000 0.8893 −9.3094 100.0000 7 −2.5621 2.9795 100.0000 2.1054 −9.9837 100.0000 8 −1.7428 3.7710 100.0000 3.3634 −10.5761 100.0000 9 −0.8676 4.5007 100.0000 4.6596 −11.0795 100.0000 10 0.0193 5.2163 100.0000 5.9884 −11.4889 100.0000 11 0.9007 5.9387 100.0000 7.3431 −11.8022 100.0000 12 1.7750 6.6696 100.0000 8.7169 −12.0166 100.0000 13 2.6425 7.4086 100.0000 10.1027 −12.1307 100.0000 14 3.5034 8.1553 100.0000 11.4931 −12.1454 100.0000 15 4.3578 8.9094 100.0000 12.8811 −12.0621 100.0000 16 5.2057 9.6707 100.0000 14.2601 −11.8840 100.0000 17 6.0473 10.4390 100.0000 15.6245 −11.6156 100.0000 18 6.8827 11.2142 100.0000 16.9691 −11.2611 100.0000 19 7.7116 11.9962 100.0000 18.2898 −10.8261 100.0000 20 8.5341 12.7850 100.0000 19.5834 −10.3159 100.0000 21 9.3500 13.5806 100.0000 20.8473 −9.7359 100.0000 22 10.1593 14.3829 100.0000 22.0793 −9.0910 100.0000 23 10.9623 15.1915 100.0000 23.2782 −8.3864 100.0000 24 11.7591 16.0062 100.0000 24.4431 −7.6267 100.0000 25 12.5500 16.8267 100.0000 25.5733 −6.8166 100.0000 26 13.3350 17.6528 100.0000 26.6688 −5.9598 100.0000 27 14.1141 18.4845 100.0000 27.7293 −5.0603 100.0000 28 14.8873 19.3216 100.0000 28.7551 −4.1213 100.0000 29 15.6547 20.1641 100.0000 29.7464 −3.1459 100.0000 30 16.4164 21.0117 100.0000 30.7037 −2.1371 100.0000 31 17.1724 21.8644 100.0000 31.6273 −1.0974 100.0000 32 17.9228 22.7221 100.0000 32.5175 −0.0290 100.0000 33 18.6675 23.5847 100.0000 33.3751 1.0659 100.0000 34 19.4067 24.4520 100.0000 34.2002 2.1854 100.0000 35 20.1403 25.3241 100.0000 34.9933 3.3277 100.0000 36 20.8684 26.2007 100.0000 35.7552 4.4912 100.0000 37 21.5911 27.0819 100.0000 36.4867 5.6740 100.0000 38 22.3083 27.9674 100.0000 37.1884 6.8747 100.0000 39 23.0202 28.8573 100.0000 37.8615 8.0917 100.0000 40 23.7267 29.7515 100.0000 38.5076 9.3233 100.0000 41 24.4280 30.6497 100.0000 39.1289 10.5675 100.0000 42 25.1240 31.5521 100.0000 39.7269 11.8231 100.0000 43 25.8149 32.4583 100.0000 40.3032 13.0888 100.0000 44 26.5008 33.3685 100.0000 40.8595 14.3635 100.0000 45 27.1815 34.2824 100.0000 41.3969 15.6462 100.0000 46 27.8573 35.2000 100.0000 41.9166 16.9362 100.0000 47 28.5280 36.1213 100.0000 42.4198 18.2327 100.0000 48 29.1939 37.0461 100.0000 42.9076 19.5351 100.0000 49 29.8549 37.9744 100.0000 43.3814 20.8427 100.0000 50 30.5111 38.9061 100.0000 43.8421 22.1549 100.0000 51 31.1625 39.8412 100.0000 44.2905 23.4714 100.0000 52 31.8092 40.7795 100.0000 44.7275 24.7917 100.0000 53 32.4511 41.7211 100.0000 45.1537 26.1156 100.0000 54 33.0885 42.6658 100.0000 45.5698 27.4426 100.0000 55 33.7211 43.6136 100.0000 45.9766 28.7725 100.0000 56 34.3492 44.5645 100.0000 46.3748 30.1051 100.0000 57 34.9727 45.5184 100.0000 46.7648 31.4400 100.0000 58 35.5916 46.4753 100.0000 47.1471 32.7772 100.0000 59 36.2060 47.4351 100.0000 47.5223 34.1164 100.0000 60 36.8159 48.3977 100.0000 47.8908 35.4575 100.0000 61 37.4214 49.3631 100.0000 48.2531 36.8002 100.0000 62 38.0225 50.3313 100.0000 48.6095 38.1445 100.0000 63 38.6192 51.3022 100.0000 48.9604 39.4903 100.0000 64 39.2117 52.2757 100.0000 49.3061 40.8374 100.0000 65 39.7999 53.2517 100.0000 49.6469 42.1857 100.0000 66 40.3839 54.2303 100.0000 49.9832 43.5352 100.0000 67 40.9637 55.2114 100.0000 50.3153 44.8858 100.0000 68 41.5394 56.1948 100.0000 50.6433 46.2373 100.0000 69 42.1110 57.1807 100.0000 50.9675 47.5897 100.0000 70 42.6786 58.1689 100.0000 51.2882 48.9430 100.0000 71 43.2421 59.1594 100.0000 51.6055 50.2971 100.0000 72 43.8017 60.1521 100.0000 51.9197 51.6519 100.0000 73 44.3574 61.1471 100.0000 52.2310 53.0074 100.0000 74 44.9092 62.1442 100.0000 52.5396 54.3634 100.0000 75 45.4572 63.1434 100.0000 52.8456 55.7201 100.0000 76 46.0013 64.1446 100.0000 53.1492 57.0773 100.0000 77 46.5417 65.1479 100.0000 53.4506 58.4351 100.0000 78 47.0784 66.1532 100.0000 53.7498 59.7932 100.0000 79 47.6115 67.1605 100.0000 54.0471 61.1519 100.0000 80 48.1410 68.1696 100.0000 54.3425 62.5109 100.0000 81 48.6669 69.1806 100.0000 54.6363 63.8703 100.0000 82 49.1893 70.1934 100.0000 54.9284 65.2300 100.0000 83 49.7082 71.2080 100.0000 55.2190 66.5901 100.0000 84 50.2236 72.2244 100.0000 55.5083 67.9504 100.0000 85 50.7356 73.2424 100.0000 55.7963 69.3110 100.0000 86 51.2443 74.2622 100.0000 56.0832 70.6719 100.0000 87 51.7496 75.2836 100.0000 56.3690 72.0329 100.0000 88 52.2517 76.3066 100.0000 56.6539 73.3942 100.0000 89 52.7507 77.3312 100.0000 56.9379 74.7556 100.0000 90 53.2465 78.3573 100.0000 57.2212 76.1172 100.0000 91 53.7392 79.3849 100.0000 57.5038 77.4790 100.0000 92 54.2288 80.4139 100.0000 57.7857 78.8409 100.0000 93 54.7153 81.4444 100.0000 58.0671 80.2029 100.0000 94 55.1988 82.4764 100.0000 58.3480 81.5650 100.0000 95 55.6792 83.5098 100.0000 58.6284 82.9272 100.0000 96 56.1567 84.5445 100.0000 58.9086 84.2894 100.0000 97 56.6315 85.5805 100.0000 59.1885 85.6517 100.0000 98 57.1035 86.6177 100.0000 59.4376 87.0179 100.0000 99 57.6014 87.6405 100.0000 58.6168 88.0148 100.0000 100 58.6168 88.0148 100.0000

It will also be appreciated that the airfoil 52 disclosed in the above Table I may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table I may be scaled upwardly or downwardly such that the airfoil profile shape remains unchanged. A scaled version of the coordinates in Table I would be represented by X, Y and Z coordinate values of Table I, with the X, Y and Z non-dimensional coordinate values converted to inches, multiplied or divided by a constant number.

An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an airfoil surface and the ideal position listed in Table I. The actual profile on a manufactured turbine nozzle will be different than those in Table I and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. As noted above, a +or −5% profile tolerance is used herein. The X, Y and Z values are all non-dimensionalized relative to the airfoil height.

The disclosed airfoil shape optimizes and is specific to the machine conditions and specifications. It provides a unique profile to achieve 1) interaction between other stages in the turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical nozzle or airfoil loadings. The disclosed loci of points defined in Table I allow the gas turbine 12 or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, any scale of the disclosed airfoil 52 may be adopted as long as 1) interaction between other stages in the pressure turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical nozzle or airfoil loadings are maintained in the scaled turbine.

The airfoil 52 described herein thus improves overall gas turbine 12 efficiency. Specifically, the airfoil 52 provides a desired turbine efficiency lapse rate (ISO, hot, cold, part load, etc.). The airfoil 52 also meets all aeromechanics and stress requirements. The turbine nozzle 50 described herein has very specific aerodynamic design requirements such as an upstream bucket radial back pressure (i.e., work splits) and radial velocity triangles for the downstream turbine rotors 150, 250, 350 and 450. Significant cross-functional design effort was require to meet these design goals. The airfoil 52 of the turbine nozzle 50 thus is of a specific shape to meet aerodynamic, mechanical, and heat transfer requirements in an efficient and cost effective manner.

This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims. 

What is claimed is:
 1. A turbine nozzle comprising an airfoil shape, the airfoil shape having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.
 2. The turbine nozzle of claim 1, wherein the turbine nozzle forms part of a stage of a turbine.
 3. The turbine nozzle of claim 1, wherein the turbine nozzle is a first stage turbine nozzle.
 4. The turbine nozzle of claim 1, wherein the airfoil shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 5. The turbine nozzle of claim 1, wherein a height of the turbine nozzle is about 5 inches to about 50 inches (about 12 centimeters to about 127 centimeters) in length.
 6. The turbine nozzle of claim 1, wherein the X, Y and Z distances are scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 7. A turbine nozzle comprising a an airfoil having a suction-side uncoated nominal airfoil profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape, the X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 8. The turbine nozzle of claim 7, wherein the turbine nozzle forms part of a stage of a turbine.
 9. The turbine nozzle of claim 7, wherein the turbine nozzle is a first stage turbine nozzle.
 10. The turbine nozzle of claim 7, wherein the airfoil shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 11. The turbine nozzle of claim 7, wherein a height of the turbine nozzle is about 5 inches to about 50 inches (about 12 centimeters to about 130 centimeters) in length.
 12. A turbine comprising a plurality of turbine nozzles, each of the turbine nozzles comprising an airfoil having an airfoil shape, the airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.
 13. The turbine of claim 12, wherein the plurality of turbine nozzles forms part of a stage of a turbine.
 14. The turbine of claim 12, wherein the plurality of turbine nozzles comprises a plurality of first stage turbine nozzles.
 15. The turbine of claim 12, wherein the airfoil shape lies in an envelope within at least one of +/−5% and/or +/−5% of a chord length in a direction normal to any airfoil surface location.
 16. The turbine of claim 12, wherein a height the turbine nozzle is about 5 inches to about 50 inches (about 12 centimeters to about 127 centimeters) in length.
 17. The turbine of claim 12, wherein the X, Y and Z distances are scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 18. The turbine of claim 12, wherein X represents a distance parallel to a turbine axis of rotation. 